Ratio of average speed to rms speed

A container is filled with n molecules of gas with speeds x,2x,3x....nx.
Find the ratio of avg speed to rms speed !
Please elaborate your answer.

Answer:
[3(x+1)/2(2x+1)]^1/2

The average speed can be found by the method Gauss used when he was 10 years old (I mention this cause it's so simple a 10 year old can do it, a 10 year old genius at least).
Average speed = Sum of Speeds / Number of speeds
Sum of speeds:
S = x + 2x + 3x + ... (n-2)x + (n-1)x + nx
S = nx + (n-1x) + (n-2x) +... 3x + 2x + 1

Notice that they are written in opposite order
Adding them together:
2S = (n+1)x + (n+1)x + (n+1)x .... (n+1)x = n(n+1)x
S = n(n+1)x/2
To get the average, divide the sum by n:
Ave = (n+1)x/2

The second sum is more complicated to prove, but it is:
Sum of n^2x^2 from 1 to n = n(n+1)(2n+1)x^2/6
You might have seen this when starting to learn integration in Calculus.

To get the average, divide the sum by n:
Ave rms= (n+1)(2n+1)x^2/6
Now take the square root to get the rms speed:
rms = sqrt[(n+1)(2n+1)x^2/6]

Ave to rms ratio:
Ave/rms =
[(n+1)x/2]/{sqrt[(n+1)(2n+1)x^2/6]} =
sqrt{6(n+1)^2x^2/[4(n+1)(2n+1)x^2]} =
sqrt{3(n+1)/[2(2n+1)]}

this is kinda close to your answer.... except there are n's instead of x's